This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic ...
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This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology. Surgery theory expresses the manifold structure set in terms of the topological K-theory of vector bundles and the algebraic L-theory of quadratic forms. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading.Less

Algebraic and Geometric Surgery

Andrew Ranicki

Published in print: 2002-09-26

This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology. Surgery theory expresses the manifold structure set in terms of the topological K-theory of vector bundles and the algebraic L-theory of quadratic forms. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading.

This chapter derives techniques for obtaining exact and approximate moments of a function of random vector/matrix. A special case of the function considered is the ratio of quadratic forms, which ...
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This chapter derives techniques for obtaining exact and approximate moments of a function of random vector/matrix. A special case of the function considered is the ratio of quadratic forms, which contains a large class of econometric estimators and test statistics. Results are analysed for both the i.i.d and non-i.i.d observations, and for linear and nonlinear models.Less

Finite Sample Moments

Aman Ullah

Published in print: 2004-05-20

This chapter derives techniques for obtaining exact and approximate moments of a function of random vector/matrix. A special case of the function considered is the ratio of quadratic forms, which contains a large class of econometric estimators and test statistics. Results are analysed for both the i.i.d and non-i.i.d observations, and for linear and nonlinear models.

This chapter shows that some notions used in a one-dimensional setting can be extended to higher dimensions and vector problems if properly modified. In particular, convexity requirements for lower ...
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This chapter shows that some notions used in a one-dimensional setting can be extended to higher dimensions and vector problems if properly modified. In particular, convexity requirements for lower semicontinuity lead to the notions of quasiconvexity and polyconvexity, and homogenization formulas are proven to hold in a vector setting once the cell-problem formula is discarded, being valid only in a convex setting. New phenomena arise that are not meaningful in one dimension, as the instability of polyconvexity by homogenization and the density of isotropic quadratic functionals in all quadratic forms.Less

*SOME COMMENTS ON VECTORIAL PROBLEMS

Andrea Braides

Published in print: 2002-07-25

This chapter shows that some notions used in a one-dimensional setting can be extended to higher dimensions and vector problems if properly modified. In particular, convexity requirements for lower semicontinuity lead to the notions of quasiconvexity and polyconvexity, and homogenization formulas are proven to hold in a vector setting once the cell-problem formula is discarded, being valid only in a convex setting. New phenomena arise that are not meaningful in one dimension, as the instability of polyconvexity by homogenization and the density of isotropic quadratic functionals in all quadratic forms.

This chapter presents a few standard definitions and results about quadratic forms and polar spaces. It begins by defining a quadratic module and a quadratic space and proceeds by discussing a ...
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This chapter presents a few standard definitions and results about quadratic forms and polar spaces. It begins by defining a quadratic module and a quadratic space and proceeds by discussing a hyperbolic quadratic module and a hyperbolic quadratic space. A quadratic module is hyperbolic if it can be written as the orthogonal sum of finitely many hyperbolic planes. Hyperbolic quadratic modules are strictly non-singular and free of even rank and they remain hyperbolic under arbitrary scalar extensions. A hyperbolic quadratic space is a quadratic space that is hyperbolic as a quadratic module. The chapter also considers a split quadratic space and a round quadratic space, along with the splitting extension and splitting field of of a quadratic space.Less

Quadratic Forms

Bernhard M¨uhlherrHolger P. PeterssonRichard M. Weiss

Published in print: 2015-09-15

This chapter presents a few standard definitions and results about quadratic forms and polar spaces. It begins by defining a quadratic module and a quadratic space and proceeds by discussing a hyperbolic quadratic module and a hyperbolic quadratic space. A quadratic module is hyperbolic if it can be written as the orthogonal sum of finitely many hyperbolic planes. Hyperbolic quadratic modules are strictly non-singular and free of even rank and they remain hyperbolic under arbitrary scalar extensions. A hyperbolic quadratic space is a quadratic space that is hyperbolic as a quadratic module. The chapter also considers a split quadratic space and a round quadratic space, along with the splitting extension and splitting field of of a quadratic space.

This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary ...
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This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary characteristic which is complete with respect to a discrete valuation ν‎ and uses the usual convention that ν‎(0) = infinity. The chapter starts with a notation regarding the ring of integers of K and the natural map from it to the residue field, followed by a number of propositions regarding an anisotropic quadratic space. These include an anisotropic quadratic space with residual quadratic spaces, an unramified quadratic space of finite dimension, unramified finite-dimensional anisotropic quadratic forms over K, unramified anisotropic quadratic forms and a bilinear form, and a round quadratic space over K. The chapter concludes with a theorem that there exists an anisotropic quadratic form over K, unique up to isometry, and is non-singular.Less

Quadratic Forms over a ∈ Local Field

Bernhard M¨uhlherrHolger P. PeterssonRichard M. Weiss

Published in print: 2015-09-15

This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary characteristic which is complete with respect to a discrete valuation ν‎ and uses the usual convention that ν‎(0) = infinity. The chapter starts with a notation regarding the ring of integers of K and the natural map from it to the residue field, followed by a number of propositions regarding an anisotropic quadratic space. These include an anisotropic quadratic space with residual quadratic spaces, an unramified quadratic space of finite dimension, unramified finite-dimensional anisotropic quadratic forms over K, unramified anisotropic quadratic forms and a bilinear form, and a round quadratic space over K. The chapter concludes with a theorem that there exists an anisotropic quadratic form over K, unique up to isometry, and is non-singular.

This chapter presents various results about quadratic forms of type F₄. The Moufang quadrangles of type F₄ were discovered in the course of carrying out the classification of Moufang polygons and ...
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This chapter presents various results about quadratic forms of type F₄. The Moufang quadrangles of type F₄ were discovered in the course of carrying out the classification of Moufang polygons and gave rise to the notion of a quadratic form of type F₄. The chapter begins with the notation stating that a quadratic space Λ‎ = (K, L, q) is of type F₄ if char(K) = 2, q is anisotropic and: for some separable quadratic extension E/K with norm N; for some subfield F of K containing K² viewed as a vector space over K with respect to the scalar multiplication (t, s) ↦ t²s for all (t, s) ∈ K x F; and for some α‎ ∈ F* and some β‎ ∈ K*. The chapter also considers a number of propositions regarding quadratic spaces and discrete valuations.Less

Quadratic Forms of Type F4

Bernhard M¨uhlherrHolger P. PeterssonRichard M. Weiss

Published in print: 2015-09-15

This chapter presents various results about quadratic forms of type F₄. The Moufang quadrangles of type F₄ were discovered in the course of carrying out the classification of Moufang polygons and gave rise to the notion of a quadratic form of type F₄. The chapter begins with the notation stating that a quadratic space Λ‎ = (K, L, q) is of type F₄ if char(K) = 2, q is anisotropic and: for some separable quadratic extension E/K with norm N; for some subfield F of K containing K² viewed as a vector space over K with respect to the scalar multiplication (t, s) ↦ t²s for all (t, s) ∈ K x F; and for some α‎ ∈ F* and some β‎ ∈ K*. The chapter also considers a number of propositions regarding quadratic spaces and discrete valuations.

This chapter presents various results about quadratic forms of type E⁶, E₇, and E₈. It first recalls the definition of a quadratic space Λ‎ = (K, L, q) of type Eℓ for ℓ = 6, 7 or 8. If D₁, D₂, and D₃ ...
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This chapter presents various results about quadratic forms of type E⁶, E₇, and E₈. It first recalls the definition of a quadratic space Λ‎ = (K, L, q) of type Eℓ for ℓ = 6, 7 or 8. If D₁, D₂, and D₃ are division algebras, a quadratic form of type E⁶ can be characterized as the anisotropic sum of two quadratic forms, one similar to the norm of a quaternion division algebra D over K and the other similar to the norm of a separable quadratic extension E/K such that E is a subalgebra of D over K. Also, there exist fields of arbitrary characteristic over which there exist quadratic forms of type E⁶, E₇, and E₈. The chapter also considers a number of propositions regarding quadratic spaces, including anisotropic quadratic spaces, and proves some more special properties of quadratic forms of type E₅, E⁶, E₇, and E₈.Less

Quadratic Forms of Type E6, E7 and E8

Bernhard M¨uhlherrHolger P. PeterssonRichard M. Weiss

Published in print: 2015-09-15

This chapter presents various results about quadratic forms of type E⁶, E₇, and E₈. It first recalls the definition of a quadratic space Λ‎ = (K, L, q) of type Eℓ for ℓ = 6, 7 or 8. If D₁, D₂, and D₃ are division algebras, a quadratic form of type E⁶ can be characterized as the anisotropic sum of two quadratic forms, one similar to the norm of a quaternion division algebra D over K and the other similar to the norm of a separable quadratic extension E/K such that E is a subalgebra of D over K. Also, there exist fields of arbitrary characteristic over which there exist quadratic forms of type E⁶, E₇, and E₈. The chapter also considers a number of propositions regarding quadratic spaces, including anisotropic quadratic spaces, and proves some more special properties of quadratic forms of type E₅, E⁶, E₇, and E₈.

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the ...
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This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.Less

Existence

Bernhard M¨uhlherrHolger P. PeterssonRichard M. Weiss

Published in print: 2015-09-15

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

This chapter uses degenerate quadratic forms and quadrics in Severi–Brauer variety to give a geometric description of all non-standard absolutely pseudo-simple k-groups G of minimal type with root ...
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This chapter uses degenerate quadratic forms and quadrics in Severi–Brauer variety to give a geometric description of all non-standard absolutely pseudo-simple k-groups G of minimal type with root system Bn over ks such that ZG = 1 and the Cartan k-subgroups of G are tori. It begins with an overview of the lemma and propositions for regular degenerate quadratic forms, coupled with two examples. It then considers the conformal isometry between quadratic spaces over a field, which is a linear isomorphism that respects the quadratic forms up to a nonzero scaling factor. It also introduces a proposition that provides sufficient conditions for an absolutely pseudo-simple k-group to be isomorphic to SO(q) for a regular quadratic form q. Finally, it describes all descents in terms of automorphisms of certain quadrics in Severi–Brauer varieties over k.Less

Constructions with regular degenerate quadratic forms

Brian ConradGopal Prasad

Published in print: 2015-11-10

This chapter uses degenerate quadratic forms and quadrics in Severi–Brauer variety to give a geometric description of all non-standard absolutely pseudo-simple k-groups G of minimal type with root system Bn over ks such that ZG = 1 and the Cartan k-subgroups of G are tori. It begins with an overview of the lemma and propositions for regular degenerate quadratic forms, coupled with two examples. It then considers the conformal isometry between quadratic spaces over a field, which is a linear isomorphism that respects the quadratic forms up to a nonzero scaling factor. It also introduces a proposition that provides sufficient conditions for an absolutely pseudo-simple k-group to be isomorphic to SO(q) for a regular quadratic form q. Finally, it describes all descents in terms of automorphisms of certain quadrics in Severi–Brauer varieties over k.

The projected system matrix in Krylov subspace methods consists of moments of the original system matrix with respect to the initial residual(s). This hints that Krylov subspace methods can be viewed ...
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The projected system matrix in Krylov subspace methods consists of moments of the original system matrix with respect to the initial residual(s). This hints that Krylov subspace methods can be viewed as matching moments model reduction. Through the simplified Stieltjes moment problem, orthogonal polynomials, continued fractions, and Jacobi matrices, we thus obtain the Gauss–Christoffel quadrature representation of the conjugate gradient method (CG). It is described how generalisations to the non-Hermitian case can easily be achieved using the Vorobyev method of moments. Finally, the described results and their historical roots are linked with the model reduction of large-scale dynamical systems. The chapter demonstrates the strong connection between Krylov subspace methods used in state-of-the-art numerical calculations and classical topics of analysis and approximation theory. Since moments represent very general objects, this suggests that Krylov subspace methods might have much wider applications beyond their immediate context of solving algebraic problems.Less

Matching Moments and Model Reduction View

Jörg LiesenZdenek Strakos

Published in print: 2012-10-18

The projected system matrix in Krylov subspace methods consists of moments of the original system matrix with respect to the initial residual(s). This hints that Krylov subspace methods can be viewed as matching moments model reduction. Through the simplified Stieltjes moment problem, orthogonal polynomials, continued fractions, and Jacobi matrices, we thus obtain the Gauss–Christoffel quadrature representation of the conjugate gradient method (CG). It is described how generalisations to the non-Hermitian case can easily be achieved using the Vorobyev method of moments. Finally, the described results and their historical roots are linked with the model reduction of large-scale dynamical systems. The chapter demonstrates the strong connection between Krylov subspace methods used in state-of-the-art numerical calculations and classical topics of analysis and approximation theory. Since moments represent very general objects, this suggests that Krylov subspace methods might have much wider applications beyond their immediate context of solving algebraic problems.

This chapter gives several examples, which may help the reader to work in concrete terms with Markoff numbers, Christoffel words, Markoff constants, and quadratic forms. In particular the thirteen ...
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This chapter gives several examples, which may help the reader to work in concrete terms with Markoff numbers, Christoffel words, Markoff constants, and quadratic forms. In particular the thirteen Markoff numbers <1000 are given, together with the associated mathematical objects considered before in the book:Markoff constants, Christoffel words, the associated matrices by the representation of Chapter 3, theMarkoff quadratic numbers whose expansion is given by the Christoffel word, the Markoff quadratic forms. Some results of Frobenius, Aigner, andClemens are given. In particular thematrix associated with a Christoffel word may be computed directly from its Markoff triple.Less

Numerology

Christophe Reutenauer

Published in print: 2018-11-15

This chapter gives several examples, which may help the reader to work in concrete terms with Markoff numbers, Christoffel words, Markoff constants, and quadratic forms. In particular the thirteen Markoff numbers <1000 are given, together with the associated mathematical objects considered before in the book:Markoff constants, Christoffel words, the associated matrices by the representation of Chapter 3, theMarkoff quadratic numbers whose expansion is given by the Christoffel word, the Markoff quadratic forms. Some results of Frobenius, Aigner, andClemens are given. In particular thematrix associated with a Christoffel word may be computed directly from its Markoff triple.

In this chapter, Markoff’s theorem for quadratic forms is proved. These forms are real, binary, and indefinite. The two first sections are concerned with results which are reminiscent of the work of ...
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In this chapter, Markoff’s theorem for quadratic forms is proved. These forms are real, binary, and indefinite. The two first sections are concerned with results which are reminiscent of the work of Gauss: each form is equivalent, under the action of GL2(Z), to a form having a root larger than 1, the other being between − 1 and 0. For such a form, onemay define a bi-infinite chain of forms, using the expansion into continued fractions of both roots; then the infimum of the form is equal to the infimum of the first coefficients of all these forms. In the last section,Markoff’s theoremis deduced: if three times the infimum of a formis larger than the square root of its discriminant, then the formmust be equivalent to aMarkoff form.Less

Markoff’s Theorem for Quadratic Forms

Christophe Reutenauer

Published in print: 2018-11-15

In this chapter, Markoff’s theorem for quadratic forms is proved. These forms are real, binary, and indefinite. The two first sections are concerned with results which are reminiscent of the work of Gauss: each form is equivalent, under the action of GL2(Z), to a form having a root larger than 1, the other being between − 1 and 0. For such a form, onemay define a bi-infinite chain of forms, using the expansion into continued fractions of both roots; then the infimum of the form is equal to the infimum of the first coefficients of all these forms. In the last section,Markoff’s theoremis deduced: if three times the infimum of a formis larger than the square root of its discriminant, then the formmust be equivalent to aMarkoff form.

Classical groups consist of the orthogonal, the unitary and the symplectic groups. These groups can be defined in terms of linear transformations that leave invariant certain quadratic forms. It ...
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Classical groups consist of the orthogonal, the unitary and the symplectic groups. These groups can be defined in terms of linear transformations that leave invariant certain quadratic forms. It follows that these are groups whose elements are matrices. The chapter shows that the following unified approach is possible: the classical groups are unitary groups over a field F; they are the orthogonal group for F = R, unitary group for F = C and unitary symplectic group for F = H. Using this unified approach the dimension and connectivity properties are obtained for all the classical groups in one fell swoop. I also note that several of the generalized orthogonal groups are of particular interest in Physics: the Lorentz group, the de Sitter group and the Liouville group. Biographical notes on Lorentz, de Sitter, Liouville, Maxwell and Thomas are given.Less

Classical groups

Adam M. Bincer

Published in print: 2012-10-11

Classical groups consist of the orthogonal, the unitary and the symplectic groups. These groups can be defined in terms of linear transformations that leave invariant certain quadratic forms. It follows that these are groups whose elements are matrices. The chapter shows that the following unified approach is possible: the classical groups are unitary groups over a field F; they are the orthogonal group for F = R, unitary group for F = C and unitary symplectic group for F = H. Using this unified approach the dimension and connectivity properties are obtained for all the classical groups in one fell swoop. I also note that several of the generalized orthogonal groups are of particular interest in Physics: the Lorentz group, the de Sitter group and the Liouville group. Biographical notes on Lorentz, de Sitter, Liouville, Maxwell and Thomas are given.

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = ...
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This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.Less

Linked Tori, II

Bernhard M¨uhlherrHolger P. PeterssonRichard M. Weiss

Published in print: 2015-09-15

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

This book goes further than the exploration of the general structure of pseudo-reductive groups to study the classification over an arbitrary field. An Isomorphism Theorem proved here determines the ...
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This book goes further than the exploration of the general structure of pseudo-reductive groups to study the classification over an arbitrary field. An Isomorphism Theorem proved here determines the automorphism schemes of these groups. The book also gives a Tits-Witt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudo-split forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions. The results and methods developed in this book will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic.Less

Classification of Pseudo-reductive Groups (AM-191)

Brian ConradGopal Prasad

Published in print: 2015-11-10

This book goes further than the exploration of the general structure of pseudo-reductive groups to study the classification over an arbitrary field. An Isomorphism Theorem proved here determines the automorphism schemes of these groups. The book also gives a Tits-Witt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudo-split forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions. The results and methods developed in this book will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic.

This book deals with the classification of pseudo-reductive groups. Using new techniques and constructions, it addresses a number of questions; for example, whether there are versions of the ...
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This book deals with the classification of pseudo-reductive groups. Using new techniques and constructions, it addresses a number of questions; for example, whether there are versions of the Isomorphism and Isogeny Theorems for pseudosplit pseudo-reductive groups and of the Existence Theorem for pseudosplit pseudo-simple groups; whether the automorphism functor of a pseudo-semisimple group is representable; or whether there is a Tits-style classification in the pseudo-semisimple case recovering the version due to Tits in the semisimple case. This introduction discusses the special challenges of characteristic 2 as well as root systems, exotic groups and degenerate quadratic forms, and tame central extensions. It also reviews generalized standard groups, minimal type and general structure theorem, and Galois-twisted forms and Tits classification.Less

Introduction

Brian ConradGopal Prasad

Published in print: 2015-11-10

This book deals with the classification of pseudo-reductive groups. Using new techniques and constructions, it addresses a number of questions; for example, whether there are versions of the Isomorphism and Isogeny Theorems for pseudosplit pseudo-reductive groups and of the Existence Theorem for pseudosplit pseudo-simple groups; whether the automorphism functor of a pseudo-semisimple group is representable; or whether there is a Tits-style classification in the pseudo-semisimple case recovering the version due to Tits in the semisimple case. This introduction discusses the special challenges of characteristic 2 as well as root systems, exotic groups and degenerate quadratic forms, and tame central extensions. It also reviews generalized standard groups, minimal type and general structure theorem, and Galois-twisted forms and Tits classification.

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It ...
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Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.Less

From Christoffel Words to Markoff Numbers

Christophe Reutenauer

Published in print: 2018-11-15

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.

It is demonstrated how d’Alembert’s Principle can be used as the basis for a more general mechanics – Lagrangian Mechanics. How this leads to Hamilton’s Principle (the Principle of Least Action) is ...
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It is demonstrated how d’Alembert’s Principle can be used as the basis for a more general mechanics – Lagrangian Mechanics. How this leads to Hamilton’s Principle (the Principle of Least Action) is shown mathematically and in words. It is further explained why Lagrangian Mechanics is so general, why forces of constraint may be ignored, and how external conditions lead to “curved space.” Also, it is explained why the Lagrangian, L, has the form L = T − V (where T is the kinetic energy and V is the potential energy), and why T is in “quadratic form” (T = 1/2mv2). It is shown how Noether’s Theorem leads to a more fundamental definition of energy and links the conservation of energy to the homogeneity of time. The ingenious Lagrange multipliers are explained, and also generalized forces and generalized coordinates.Less

Lagrangian Mechanics

Jennifer Coopersmith

Published in print: 2017-05-11

It is demonstrated how d’Alembert’s Principle can be used as the basis for a more general mechanics – Lagrangian Mechanics. How this leads to Hamilton’s Principle (the Principle of Least Action) is shown mathematically and in words. It is further explained why Lagrangian Mechanics is so general, why forces of constraint may be ignored, and how external conditions lead to “curved space.” Also, it is explained why the Lagrangian, L, has the form L = T − V (where T is the kinetic energy and V is the potential energy), and why T is in “quadratic form” (T = 1/2mv2). It is shown how Noether’s Theorem leads to a more fundamental definition of energy and links the conservation of energy to the homogeneity of time. The ingenious Lagrange multipliers are explained, and also generalized forces and generalized coordinates.